Steam.exe (mail exception): ERROR: copying steamNEW.exe to Steam.exe failed edit

When you got this error message when you're trying to run Steam.

Steam.exe (mail exception): ERROR: copying steamNEW.exe to Steam.exe failed, Win32 Error 32 "The process cannot access the file because it is being used by another process."

Try to run Steam.exe or SteamNew.exe as Administrator account. (right click it and click 'run as administrator'. It's in your steam directory)